A new type of reflected backward doubly stochastic differential equations

TL;DR

This paper introduces a new class of reflected backward doubly stochastic differential equations with nonlinear drift depending on the barrier, extending previous work on stochastic variants and providing foundational results for their analysis.

Contribution

It extends the concept of variant reflected BSDEs to the doubly stochastic case, establishing existence, uniqueness, and comparison theorems for these new equations.

Findings

Established the stochastic variant Skorohod problem.

Proved existence and uniqueness of solutions for VRBDSDEs.

Demonstrated comparison and stability results for solutions.

Abstract

In this paper, we introduce a new kind of "variant" reflected backward doubly stochastic differential equations (VRBDSDEs in short), where the drift is the nonlinear function of the barrier process. In the one stochastic case, this type of equations have been already studied by Ma and Wang. They called it as "variant" reflected BSDEs (VRBSDEs in short) based on the general version of the Skorohod problem recently studied by Bank and El Karoui. Among others, Ma and Wang showed that VRBSDEs is a novel tool for some problems in finance and optimal stopping problems where no existing methods can be easily applicable. Since more of those models have their stochastic counterpart, it is very useful to transpose the work of Ma and Wang to doubly stochastic version. In doing so, we firstly establish the stochastic variant Skorohod problem based on the stochastic representation theorem, which extends the work of Bank and El Karoui. We prove the existence and uniqueness of the solution for VRBDSDEs by means of the contraction mapping theorem. By the way, we show the comparison theorem and stability result for the solutions of VRBDSDEs.